Heat Analysis and Thermodynamic Effects Part 10 pptx

Integral Transform Method Versus Green Function Method in Electron, Hadron or Laser Beam - Water Phantom Interaction 259    2 0 ij ij ij Kx kx K x xx          (4) Equation (4) provides a series of positive eigenvalues , j jN   and eigenfunctions  , ij i Kx of the differential operator:  kx xx        , [7]. The eigenfunctions were necessary for solving equation (2). The solution of equation (4) has the form:      00 22 ,, ji i i ji i i ij ii i ii k x mx k x mx Kx AJ BY j N mm            (5) Where: iii c and J 0 and Y 0 are the Bessel and Weber functions respectively. After the application of the integral operator K i (x) equation (4) becomes:     2 , ,, i j i jj ii j i Tt utc f t t          (6) where:    1 1 ,(,), i i x i i jj j x ut TxtKxdx C          1 1 ,,, i i x j jj j x f t f xtK x dx C      and  j C  is a normalization factor. Here we have [7]: 1 1 2 0 () (,) i i x n jijij i x CKxdx        . (7) In the same manner, one can apply the functions: (,) kk Ky and (,) ll Kz , which satisfy the equations:      2 2 2 2 2 2 , ,0 , ,0 kk kk k ll lll Ky Ky y Kz Kz z          (8) This next gives: (,) cos( )(/ )sin( ) ll l l l Kz zhk z    , with, k being the thermal conductivity and h the heat transfer coefficient . Heat Analysis and Thermodynamic Effects 260 Then, the following equation is inferred: 222 ˆˆˆ ( , ,,) (, ,,) ( , ,,) ˆ ˆ (,,,) (,,,) ji j k l ki j k l li j k l ij kl i ii TtTtTt Tt fxyzt c t            (9) Where from it follows: 2 13 23 1 ()()() /2 /2 /2 /2 ˆ (,,,) (,,,) ( ,) ( ,) ( ,) jkl i i ijkl CCC y xz ij j k k l l xy z Tt T x y z t K x K y K z dxdydz          (10) In order to eliminate the time parameter t, we apply the direct and inverse Laplace transform to equation (9). If we have, like in most cases: 0 (,,,) (,,)[() ( )]fxyzt fxyz ht ht t   , one can get the solution: 22 2 22 2 0 22 2 () ()() 1 0 () 111 (,,,) [1 (1 )( )] (,,) (,) (,) (,) jl k jl k jl k ttt i jkl jkl ijj kk ll Txyzt e e ht t gKxKyKz                       (11) where: 2 13 23 1 ()()() /2 /2 1 0 /2 /2 (,,) (,,,) ( ,) ( ,) ( ,) jkl i i jkl CCC y xz n iijjkkll i xy z g f x y z t K x K y K z dxdydz              (12)  stands here for the thermal diffusivity. We point out that our semi-analytical solution becomes analytical, if we observe the, after 10 iterations, the solution becomes convergent (we have values of temperature less than 10 -2 K for: i>10;j>10 and k>10). Under these conditions equation (10) becomes: 22 2 22 2 0 22 2 10 10 10 () ()() 1 0 () 111 (,,,) [1 (1 )( )] (,,) (,) (,) (,) jl k jl k jl k ttt i jkl jkl ijj kk ll Txyzt e e ht t gKxKyKz                      (13) 3. Application of the theory: Laser-assisted hadron and electron beams therapy It is well known [8] that the hadrons therapy (e.g. with protons) is much more suitable and efficient compared to electrons therapy, because the absorption curve is (in this case) a Dirac function. We can solve easily the heat equation for this case, and we obtain the temperature field in Fig.1. - (The Dirac absorption function is at 4 cm from the surface). Here dT is the temperature variation (dT=T f -T i ), rather than the absolute temperature. Integral Transform Method Versus Green Function Method in Electron, Hadron or Laser Beam - Water Phantom Interaction 261 T f and T i are the final and initial temperatures respectively. Fig. 1. Thermal field distribution in case of 1 MeV proton beam irradiation of a water phantom, for 120 sec. Fig. 2. Thermal field in water submitted to cw CO 2 laser irradiation for 50 sec. Heat Analysis and Thermodynamic Effects 262 The power of the cw CO 2 laser beam was P= 1W. It is known from experience [8] that proton therapy is more efficient in the “presence” of a laser beam. We plotted in figure 2 the thermal gradient in water produced by cw CO 2 laser irradiation for 50 sec. (P = 1W). In Fig. 3 we presented the temperature field in water produced by an electron beam, when the “steady - state” is achieved. The white color corresponds to an increase of temperature, and the black color represents a decrease of temperature. We have use sub-domains of 0.25 cm. The thickness of the water phantom was 0.25 cm, and was contained in a plastic cube with a mass density close to 1 g/cm 3 . Figure 3 was obtained using eq. (13). Fig. 3. Temperature field in water produce by an electron beam, when the “steady- state” is achieved. The white color corresponds to the temperature increase while, the black color represents the temperature decrease. We have used sub-domains of 0.25 cm length. 4. The green function method We start from the heat equation:   [()] [()] [()] (,,) TTT xyz xXyyzz KT KT KT Sx y z     (14) where S(x, y, z) is proportional with the absorbed dose. We consider [9], the case of a 10 MeV electron beam interactions with water. We have: 10 (,,) (,). ()Sxyz Kyz D x  (15) where according to experimental data from our laboratory: 2345 10 ( ) 83.2337 18.6522 15.1080 4.1417 0.3506Dx x x x x   (16) Integral Transform Method Versus Green Function Method in Electron, Hadron or Laser Beam - Water Phantom Interaction 263 Here x stands for the direction of electron propagation. We will consider the radiation (electron beam) normal to water surface. From the standard theory of Green function applied to multi-layer structures, we have: 12 12 12 12 12 n ll ll ll nn kk k kk k nn ll l l K         (17) where l i is the length and k i is the thermal conductivity of the i-th layer. We introduce the area of the layer A i : 11 22 11 22 12 nn nn n KA KA KA KA KA KA AA A A K        (18) We define the “linear” temperature 0 00 () ( ) (1/( )) (') ' T T T T KT KT dT   (19) and we can write: 3 1 22 0 [1 ( )] () () ( ) 0 () PRT KT f d         (20) where: K K   and: ()KKT (21) The function f  is given by:  22 2 2 2 2 2 exp [[ ( 1)] [ /( 1)] ( / )] (1) () XY Z f      (22) We plotted in Fig.4 the analytical results obtained with the Green function method. The white color corresponds to temperature increase, and the black color represents a decrease of temperature. We have used sub-domains of 0.25 cm length. Figs. 3 and 4 allow for a direct comparison between the temperature fields in water computed with the integral transform technique and Green function method under identical conditions. 5. The thermal fields when we have multiple sources irradiations We consider a parallelepiped sample with dimensions a, b, and c. The sample is irradiated by three laser beams which propagate along the Cartesian coordinate axes. The model is also valid for electron or hadrons beam irradiations. Heat Analysis and Thermodynamic Effects 264 Let us considering the following relations: 123 (,,,) (,,,) (,,,) (,,,)Axyzt A xyzt A xyzt A xyzt (23) Therefore: 123 (,,,) (,,,) (,,,) (,,,)Txyzt T xyzt T xyzt T xyzt (24) Fig. 4. The temperature field in water produced by a 10 MeV electron beam, when the “steady- state” is achieved. We suppose that for the heat transfer coefficients: 123456 hhhhhhh   . If we consider a linear heat transfer at the sample surface (the “radiation” boundary condition [11]), we have: for the first laser beam , direction of propagation along x axis: 22 2 22 2 0; 0; 0; 0; 0; 0 y xx xxy aa b xx y y zz yzz cc b zz y K KK hhh KKK xK xK yK K KKhhh KKK yK zK zK                                   (25) for the second laser beam, direction of propagation along y axis: Integral Transform Method Versus Green Function Method in Electron, Hadron or Laser Beam - Water Phantom Interaction 265 22 2 22 2 0; 0; 0; 0; 0; 0 y xx xxy aa b xx y y zz yzz cc b zz y L LL hhh LLL xK xK yK L LL hhh LLL yK zK zK                                (26) for the third laser beam, direction of propagation along z axis: 22 2 22 2 0; 0; 0; 0; 0; 0 y xx xxy aa b xx y y zz yzz cc b zz y M MM hhh MMM xK xK yK M MM hhh MMM yK zK zK                             (27) The solution of the heat equation subjected to boundary conditions (25), (26) and (27) is: , 111 ,111 (,,,) (,,)(,,,)(,)(,)(,) (,,)(,,,)(,)(,)(,) (, , )(, , ,) (,) ijo ijo xi yj zo mn i j o pr s vr s xv yr zs vprs tvw tvw xt Txyzt a b tK xK yK z cdtLxLyLz eftMx                     ,111 (,) ( ,) yv zw rtvw My Mz               (28) We have:  22 2 2 2 2 0 (,) ( ) ( ) exp yxy x mn mn m n ww w IxyI H H               (29) Here w is the width of the laser beam. 0, , , 2 2 22 , 22 (,,) ( (1 )(,) ()) ( ,) ( ,) i xmn a mn x ijo i Sxi a ijo bc SS mnyj zo bc I aerKx KC C C rxdx IKyKzdydz                   (30) where: 22 () 222 1 (,,,) [1 (1 )( )] ijo ijo o ttt ijo o ijo bt eehtt           (31) and Heat Analysis and Thermodynamic Effects 266 2222 () ijo i j o     . We have: (,)cos( )(/ )sin( ) xi i i i Kx xhk x    (32) The other formulas can be easily obtains by “rotations” of the indices. t -is the time and o t the exposure time. We have: S r is the parameter which take care of the surface absorption and which make sense only for one photon absorption. Here : 222 ,,,,, ii pp tt     are the eigenvalues corresponding to the eigenfunctions: ,, ,,,,, ,,,,,, , , ,,, xyzxyzxyzxyz x y z x y z KKKPPPLLLTTTMMMNNN [7]. ,,,,,, ij o p r s t v CC C C C C C C and w C as well similarly formulas for two photon absorption. 0 ()ht t is the step function [7]. We can generalize formula (28) taking into account the one and two absorption coefficient. In this case we have the following solution: ,111 222 222 2 2 2 ,111 (,,,) ( , , )( , , ,) ( ,) ( ,) ( ,) (,,)(,,,)(,)(,)(,) (,,)(,,, ijo ijo xi yj zo mn i j o ijo ijo xi yj zo mn i j o pr s vr s Txyzt a b tK xK yK z btPxPyPz cdt                     ,111 222 22 2 2 2 2 ,111 11 )(,)(,)(,) (,,)(,,,)(,)(,)(,) (, , )(, , ,) (,) ( ,) ( ,) xv yr zs vq p r s pr s vr s xv yr zs vq p r s tvw tvw xt yv zw vw LxLyLz dtTxTyTz eftMxMyMz                       ,1 22 2 22 2 2 2 2 ,111 (, , )(, , ,)(,)( ,)( ,) rt tvw tvw xt yv zw rtvw f tN xN y Nz                       (33) In formula (33) the upper index 2 means that the corresponding values are connected with two photon absorption. The eigenfunctions and the eigenvalues for two absorption phenomena can be calculated in the same way like in the case of one photon absorption with the only change that we have another absorption formula. It make no sense to take into account three or more photons absorption phenomena because in this situations the cross sections are very small. In the next pages we will present three simulations, using the developed “multiple beam irradiation”. The different characteristics of dielectrics under one laser beam irradiation have been very well studied in literature. We will take the case of a ZnSe sample (all characteristics of the material can be found in reference [11]). The sample is a cube with the dimension about 2 cm. Integral Transform Method Versus Green Function Method in Electron, Hadron or Laser Beam - Water Phantom Interaction 267 Fig. 5. Temperature field in the plane x=0, during a 100s irradiation with a 10 W CO 2 laser beam. Fig. 6. Temperature field plotted during 100s irradiation with a 50 W CO 2 laser beam, operating in the TEM 03 . Heat Analysis and Thermodynamic Effects 268 Our study indicates that for a sample under one, two or three laser irradiation, the heat equation has an exact semi-analytical solution. In fact it can be considered an analytical solution because the eigenvalues with index higher than 10 does not contribute to the solution of heat equation. This solution it is not simply the sum of solutions from three one- dimensional heat equations, because 12 (, ,,), (, ,,) TxyztT xyzt and 3 (, ,,)Txyzt are coupled via boundary conditions. Our model can be easily generalized for the cases when: 123456 hhhh hhor   x yz   .The model could be applied to any laser-solid system whose interaction can be described by Beer law. The integral transform technique has proved once again it’s “power” in resolving heat equation problems [14-17]. Fig. 7. Temperature field when the sample is irradiated simultaneously with the two laser beams, mentioned above (Fig.5 and Fig.6) 6. Discussions and conclusions We developed a method for solving the heat diffusion equation- based on dividing the whole domain into small intervals, the length of each depending on the required accuracy of the final solution. The theory is applicable to laser, electrons and hadrons beams interaction with human tissues (which are simulated by a water phantom). In each of the obtained intervals the thermal conductivity function is approximated by a linear function. This function is introduced in the heat equation associated to each interval. At the interface between intervals, the continuity of temperature function and its first derivative are ensured, these conditions providing the values for the coefficients obtained in the final solution. [...]... 2009 270 Heat Analysis and Thermodynamic Effects 8 References [1] A P Kubyshkin, M P Matrosov, and A A Karabutov, Opt Eng., 3214, 1996, [2] M D Dramicanin, Z D Ristovski, V Djokovic, and S Galovic: Appl Phys Lett 73, 321, 1998, [3] J Opsal and A Rosencwaig: J Appl Phys 53, 6, 4240, 1982, [4] Z Bozóki , A Miklós, and D Bicanic , Appl Phys Lett 64, 11, 1362, 1994, [5] S Bhattacharyya, A Pal, and A S Gupta:... MEMS technology and was composed of lower, middle, and upper substrates The lower substrate was composed of silicon, while the middle and upper substrates were made from Pyrex glass for visualization Through a preliminary test, it was verified that there was no leakage at the adhesion interface between the lower and the middle or upper substrates 272 Heat Analysis and Thermodynamic Effects and at the bonding... Vol.51, R 455, 2006, [10] M Oane, I Morjan, and R Medianu, Optics and Laser Technology, 36, 677, 2004, [11] M Oane and D Sporea, Infrared Physics & Technology, 42, 31, 2001, [12] M Oane and D Apostol, Optics and Laser Technology, 36, 219, 2004, [13] M Oane, S L Tsao, and F Scarlat, Optics and Laser Technology, 39, 179, 2007, [14] M Oane, A Peled, Fl Scarlat, I N Mihailescu, A Scarisoreanu, and G Georgescu,... albeit small in amount, can be detrimental to the heat pipe performance 278 Heat Analysis and Thermodynamic Effects Fig 7 shows the temperature distribution by the axial length of 50 mm The tested MHP has a curved triangular cross section and a 20% filling ratio to the inner total volume The heat was dissipated only at the condenser with the conduction heat transfer The temperatures were averaged over... comparison by cross-section type 6 7 8 282 Heat Analysis and Thermodynamic Effects 10 o Tv=60 C o Tv=70 C o Tv=80 C o Tv=90 C o Thermal Resistance ( C/W) 8 Dryout points 6 4 2 0 0 2 4 6 8 10 12 Input Power (W) Fig 12 Thermal performance by the operating temperature 12 Experimental data(Present study) Experimental data(Moon et al,1999) Heat Transfer Limit (W) 10 8 6 4 2 50 60 70 80 90 o Operating Temperature... electronic and communication devices to become thinner and thinner, flat plate type cooling devices offer great convenience to be applied to such devices compared with circular type cooling devices In the present, the flat heat pipe pressed from a circular shape has been widely used in notebook PCs, sub-notebook PCs, 284 Maximum Heat Transport Capacity (W) Heat Analysis and Thermodynamic Effects Copper... of 280 Heat Analysis and Thermodynamic Effects Moon et al over the operating temperature of 60–80 °C This result shows that large capillary limit was obtained in the present study compared to that in Moon (Moon et al., 1999) High productivity and simple manufacturing process were considered, and enhanced performance was obtained compared to that of Moon et al for the future applications Figs 14 and 15... 20% Heat Transfer Limit (W) 20 15 10 5 0 -90 -45 0 45 90 Inclination Angle (degree) Fig 9 Thermal performance by inclination angle 2o Overall Heat Transfer Coefficient (W/m C) 1400 Ltotal =100 mm Ltotal=50mm 1200 100 0 800 600 400 200 0 0 1 2 3 4 Input Power (W) Fig 10 Overall heat transfer coefficient by total length Curved rectangular MHP Curved triangular MHP 20 o Thermal Resistance ( C/W) 25 15 10. .. operating temperature Maximum Heat Transport Capacity (W) 1.4 Copper Percent of Filling : 30% 1.2 1.0 0.8 0.6 0.4 0.2 0.0 30 40 50 60 70 80 90 o Operating Temperature( C) Fig 1 Maximum heat transport capacity according to operating temperatures 274 Heat Analysis and Thermodynamic Effects 0.0007 0.0006 Capillary radius (m) 0.0005 0.0004 0.0003 0.0002 0.0001 0.0000 0.000 0.005 0. 010 0.015 0.020 0.025 0.030... minimized, but their performance is becoming better and better Their heat flux has been significantly increased and has already exceeded about 100 W/cm2 recently The insufficient dissipating of the heat flux may lead to performance decrease or failure of the electronic device and components Heat flux in laptop computers has not been questioned; therefore, only a heat sink has been applied on cooling them Recently, . have values of temperature less than 10 -2 K for: i> ;10; j> ;10 and k> ;10) . Under these conditions equation (10) becomes: 22 2 22 2 0 22 2 10 10 10 () ()() 1 0 () 111 (,,,) [1 (1 )( )] . interface between the lower and the middle or upper substrates Heat Analysis and Thermodynamic Effects 272 and at the bonding interface between the lower substrate and the fill tube. Although. a 100 s irradiation with a 10 W CO 2 laser beam. Fig. 6. Temperature field plotted during 100 s irradiation with a 50 W CO 2 laser beam, operating in the TEM 03 . Heat Analysis and Thermodynamic